Respuesta :

[tex]\bf \textit{Sum and Difference Identities} \\\\ sin(\alpha - \beta)=sin(\alpha)cos(\beta)- cos(\alpha)sin(\beta) \\\\ cos(\alpha - \beta)= cos(\alpha)cos(\beta) + sin(\alpha)sin(\beta)\\\\ -------------------------------\\\\ cot\left(x-\frac{\pi }{2} \right)=-tan(x)\\\\ -------------------------------[/tex]

[tex]\bf cot\left(x-\frac{\pi }{2} \right)\implies \cfrac{cos\left(x-\frac{\pi }{2} \right)}{sin\left(x-\frac{\pi }{2} \right)}\implies \cfrac{cos(x)cos\left( \frac{\pi }{2} \right)+sin(x)sin\left( \frac{\pi }{2} \right)}{sin(x)cos\left( \frac{\pi }{2} \right)-cos(x)sin\left( \frac{\pi }{2} \right)} \\\\\\ \cfrac{cos(x)\boxed{0}+sin(x)\boxed{1}}{sin(x)\boxed{0}-cos(x)\boxed{1}}\implies \cfrac{sin(x)}{-cos(x)}\implies -tan(x)[/tex]
Cetacea

We have shown that: [tex]\cot\left(x-\frac{\pi}{2}\right)=-\tan x[/tex]

We will use the formula: cot(-x) = -cot(x) and [tex]\cot\left(\frac{\pi}{2}-x\right)=\tan x[/tex]

Using the formulas we get:

[tex]\cot\left(x-\frac{\pi}{2}\right)=\cot\left(-\left(\frac{\pi}{2}-x\right)\right)=-\cot\left(\frac{\pi}{2}-x\right)=-\tan x[/tex].

Learn more: https://brainly.com/question/10241002

Otras preguntas